I'd wind the scope like 30 clicks left, 60 clicks right, 30 click back left. Same with the elevation (up down, obviously).
Then try and fine tune the zero again. Quite likely this will cause a microscopic change in where the zero settles, which is all you are looking for.

Otherwise just loosen the mount clamps, and re-tighten (making no conscious changes). Same thing - you'll get a microscopic change.

In reality, I never been able to shoot good enough for me to notice 1 click, unless my scope was jumping around a bit (i.e. not true 1/4" at 100 yards), but if you are benching with a quality setup, I guess it's entirely possible.

Shimming is a waste of time, don't even think about it. All the other changes you are introducing will be more than enough on their own.

I'd also be very suspicous of other factors at play here, e.g paralax error, or cant. When shooting your "fine tuning" group, make a point of de-resting your rifle, and re-composing your shooting position (as you would between groups) - between each pellet shot. that will show up any hold / paralax / eye positioning / cant issues.

HTH - JB